2-ancestry mixing, negative discriminant

Percentage Accurate: 98.5% → 100.0%

Time: 11.9s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))

C

double code(double g, double h) {
	return 2.0 * cos((((2.0 * ((double) M_PI)) / 3.0) + (acos((-g / h)) / 3.0)));
}

Java

public static double code(double g, double h) {
	return 2.0 * Math.cos((((2.0 * Math.PI) / 3.0) + (Math.acos((-g / h)) / 3.0)));
}

Python

def code(g, h):
	return 2.0 * math.cos((((2.0 * math.pi) / 3.0) + (math.acos((-g / h)) / 3.0)))

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(Float64(Float64(2.0 * pi) / 3.0) + Float64(acos(Float64(Float64(-g) / h)) / 3.0))))
end

MATLAB

function tmp = code(g, h)
	tmp = 2.0 * cos((((2.0 * pi) / 3.0) + (acos((-g / h)) / 3.0)));
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[(2.0 * Pi), $MachinePrecision] / 3.0), $MachinePrecision] + N[(N[ArcCos[N[((-g) / h), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(\sqrt{3}, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi}, -\cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (-
  (fma
   (sqrt 3.0)
   (sin (* 0.3333333333333333 (acos (- (/ g h)))))
   (cos (* 0.3333333333333333 (fma (sqrt PI) (sqrt PI) (- (acos (/ g h)))))))))

C

double code(double g, double h) {
	return -fma(sqrt(3.0), sin((0.3333333333333333 * acos(-(g / h)))), cos((0.3333333333333333 * fma(sqrt(((double) M_PI)), sqrt(((double) M_PI)), -acos((g / h))))));
}

Julia

function code(g, h)
	return Float64(-fma(sqrt(3.0), sin(Float64(0.3333333333333333 * acos(Float64(-Float64(g / h))))), cos(Float64(0.3333333333333333 * fma(sqrt(pi), sqrt(pi), Float64(-acos(Float64(g / h))))))))
end

Wolfram

code[g_, h_] := (-N[(N[Sqrt[3.0], $MachinePrecision] * N[Sin[N[(0.3333333333333333 * N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(0.3333333333333333 * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision] + (-N[ArcCos[N[(g / h), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}} \]

6. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \left(\sqrt{3} \cdot -0.5\right) \cdot \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}}} \]

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right)\right) \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \frac{-1}{2}\right)} \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   3. neg-mul-1 N/A

      \[\leadsto \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(-1 \cdot \frac{g}{h}\right)}\right) \cdot \frac{-1}{2}\right) \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \left(\frac{-1}{2} \cdot 2\right)} + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \color{blue}{-1} + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot -1 + \color{blue}{\left(\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right) \cdot \frac{-1}{2}\right)} \cdot 2 \]

   7. neg-mul-1 N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot -1 + \left(\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(-1 \cdot \frac{g}{h}\right)}\right) \cdot \sqrt{3}\right) \cdot \frac{-1}{2}\right) \cdot 2 \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\cos^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{g}{h}}\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    2. acos-neg N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) - \cos^{-1} \left(\frac{g}{h}\right)\right)} \cdot \frac{1}{3}\right)\right)\right) \]

    3. sub-neg N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right)} \cdot \frac{1}{3}\right)\right)\right) \]

    4. add-sqr-sqrt N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)}, \mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right)} \cdot \frac{1}{3}\right)\right)\right) \]

    6. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}, \sqrt{\mathsf{PI}\left(\right)}, \mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    7. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\mathsf{fma}\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}, \sqrt{\mathsf{PI}\left(\right)}, \mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    8. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}, \mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    9. lower-PI.f64 N/A

       \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}, \mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)\right) \cdot \frac{1}{3}\right)\right)\right) \]

    10. lower-neg.f64 N/A

        \[\leadsto \mathsf{neg}\left(\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right) \cdot \frac{1}{3}\right), \cos \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\mathsf{neg}\left(\cos^{-1} \left(\frac{g}{h}\right)\right)}\right) \cdot \frac{1}{3}\right)\right)\right) \]

    11. lower-acos.f64 100.0

        \[\leadsto -\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi}, -\color{blue}{\cos^{-1} \left(\frac{g}{h}\right)}\right) \cdot 0.3333333333333333\right)\right) \]

11. Applied rewrites 100.0%

    \[\leadsto -\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\color{blue}{\mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi}, -\cos^{-1} \left(\frac{g}{h}\right)\right)} \cdot 0.3333333333333333\right)\right) \]

12. Final simplification 100.0%

    \[\leadsto -\mathsf{fma}\left(\sqrt{3}, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi}, -\cos^{-1} \left(\frac{g}{h}\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\\ -\mathsf{fma}\left(\sqrt{3}, \sin t\_0, \cos t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (acos (- (/ g h))))))
   (- (fma (sqrt 3.0) (sin t_0) (cos t_0)))))

C

double code(double g, double h) {
	double t_0 = 0.3333333333333333 * acos(-(g / h));
	return -fma(sqrt(3.0), sin(t_0), cos(t_0));
}

Julia

function code(g, h)
	t_0 = Float64(0.3333333333333333 * acos(Float64(-Float64(g / h))))
	return Float64(-fma(sqrt(3.0), sin(t_0), cos(t_0)))
end

Wolfram

code[g_, h_] := Block[{t$95$0 = N[(0.3333333333333333 * N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, (-N[(N[Sqrt[3.0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision] + N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing

4. Applied rewrites 99.9%

   \[\leadsto 2 \cdot \color{blue}{\frac{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{3} - {\left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right)}^{3}}{{\left(\left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}^{2} + \left(2 \cdot \left(\left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)\right) \cdot \mathsf{fma}\left(2, \left(\frac{\sqrt{3}}{2} \cdot 0.5\right) \cdot \sin \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right), \left(0.25 - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) \cdot \cos \left(\cos^{-1} \left(\frac{g}{-h}\right) \cdot 0.3333333333333333\right)\right)}} \]

6. Applied rewrites 98.4%

   \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(-0.5, \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \left(\sqrt{3} \cdot -0.5\right) \cdot \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)}}} \]

7. Taylor expanded in g around 0

   \[\leadsto \color{blue}{2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) + \frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right)} \]
8. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right)\right) \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \frac{-1}{2}\right)} \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   3. neg-mul-1 N/A

      \[\leadsto \left(\cos \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(-1 \cdot \frac{g}{h}\right)}\right) \cdot \frac{-1}{2}\right) \cdot 2 + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \left(\frac{-1}{2} \cdot 2\right)} + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   5. metadata-eval N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot \color{blue}{-1} + \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right)\right) \cdot 2 \]

   6. *-commutative N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot -1 + \color{blue}{\left(\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \left(\mathsf{neg}\left(\frac{g}{h}\right)\right)\right) \cdot \sqrt{3}\right) \cdot \frac{-1}{2}\right)} \cdot 2 \]

   7. neg-mul-1 N/A

      \[\leadsto \cos \left(\frac{1}{3} \cdot \cos^{-1} \left(-1 \cdot \frac{g}{h}\right)\right) \cdot -1 + \left(\left(\sin \left(\frac{1}{3} \cdot \cos^{-1} \color{blue}{\left(-1 \cdot \frac{g}{h}\right)}\right) \cdot \sqrt{3}\right) \cdot \frac{-1}{2}\right) \cdot 2 \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(\sqrt{3}, \sin \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right), \cos \left(\cos^{-1} \left(-\frac{g}{h}\right) \cdot 0.3333333333333333\right)\right)} \]

10. Final simplification 100.0%

    \[\leadsto -\mathsf{fma}\left(\sqrt{3}, \sin \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right), \cos \left(0.3333333333333333 \cdot \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \]
11. Add Preprocessing

Alternative 3: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (/ (fma (acos (- (/ g h))) (/ -1.5 PI) -3.0) (* 3.0 (/ -1.5 PI))))))

C

double code(double g, double h) {
	return 2.0 * cos((fma(acos(-(g / h)), (-1.5 / ((double) M_PI)), -3.0) / (3.0 * (-1.5 / ((double) M_PI)))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(fma(acos(Float64(-Float64(g / h))), Float64(-1.5 / pi), -3.0) / Float64(3.0 * Float64(-1.5 / pi)))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * N[(-1.5 / Pi), $MachinePrecision] + -3.0), $MachinePrecision] / N[(3.0 * N[(-1.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)}{3}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   6. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\color{blue}{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

   8. +-commutative N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}}\right) \]

   11. clear-num N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{1}{\frac{3}{2 \cdot \mathsf{PI}\left(\right)}}}\right) \]

   12. frac-2neg N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)}\right) \]

   14. frac-add N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) + 3 \cdot -1}{3 \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}\right)} \]

   15. lower-/.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) + 3 \cdot -1}{3 \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right)} \]

5. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right) \]
6. Add Preprocessing

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\pi \cdot \frac{\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{-4.5}\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (* PI (/ (fma (acos (- (/ g h))) (/ -1.5 PI) -3.0) -4.5)))))

C

double code(double g, double h) {
	return 2.0 * cos((((double) M_PI) * (fma(acos(-(g / h)), (-1.5 / ((double) M_PI)), -3.0) / -4.5)));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(pi * Float64(fma(acos(Float64(-Float64(g / h))), Float64(-1.5 / pi), -3.0) / -4.5))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(Pi * N[(N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * N[(-1.5 / Pi), $MachinePrecision] + -3.0), $MachinePrecision] / -4.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)}{3}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   6. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\color{blue}{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

   8. +-commutative N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}}\right) \]

   11. clear-num N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{1}{\frac{3}{2 \cdot \mathsf{PI}\left(\right)}}}\right) \]

   12. frac-2neg N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{\color{blue}{-1}}{\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)}\right) \]

   14. frac-add N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) + 3 \cdot -1}{3 \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}\right)} \]

   15. lower-/.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right) + 3 \cdot -1}{3 \cdot \left(\mathsf{neg}\left(\frac{3}{2 \cdot \mathsf{PI}\left(\right)}\right)\right)}\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), \frac{-1.5}{\pi}, -3\right)}{3 \cdot \frac{-1.5}{\pi}}\right)} \]
5. Step-by-step derivation

   1. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{\color{blue}{\mathsf{neg}\left(h\right)}}\right) \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)} + -3}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \color{blue}{\left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)} + -3}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   3. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right)} \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)} + -3}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   4. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \frac{\frac{-3}{2}}{\color{blue}{\mathsf{PI}\left(\right)}} + -3}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right) \cdot \color{blue}{\frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}} + -3}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   6. lift-fma.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}}{3 \cdot \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{3 \cdot \frac{\frac{-3}{2}}{\color{blue}{\mathsf{PI}\left(\right)}}}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{3 \cdot \color{blue}{\frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}}\right) \]

   9. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{3 \cdot \color{blue}{\frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}}}\right) \]

   10. associate-*r/ N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{\color{blue}{\frac{3 \cdot \frac{-3}{2}}{\mathsf{PI}\left(\right)}}}\right) \]

   11. associate-/r/ N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{3 \cdot \frac{-3}{2}} \cdot \mathsf{PI}\left(\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{\mathsf{neg}\left(h\right)}\right), \frac{\frac{-3}{2}}{\mathsf{PI}\left(\right)}, -3\right)}{3 \cdot \frac{-3}{2}} \cdot \mathsf{PI}\left(\right)\right)} \]

6. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{-4.5} \cdot \pi\right)} \]

7. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\pi \cdot \frac{\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), \frac{-1.5}{\pi}, -3\right)}{-4.5}\right) \]
8. Add Preprocessing

Alternative 5: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), -3, \pi \cdot -6\right) \cdot -0.1111111111111111\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (*
  2.0
  (cos (* (fma (acos (- (/ g h))) -3.0 (* PI -6.0)) -0.1111111111111111))))

C

double code(double g, double h) {
	return 2.0 * cos((fma(acos(-(g / h)), -3.0, (((double) M_PI) * -6.0)) * -0.1111111111111111));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(fma(acos(Float64(-Float64(g / h))), -3.0, Float64(pi * -6.0)) * -0.1111111111111111)))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(N[(N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision] * -3.0 + N[(Pi * -6.0), $MachinePrecision]), $MachinePrecision] * -0.1111111111111111), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)}{3}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   6. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \frac{\color{blue}{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \mathsf{PI}\left(\right)}{3} + \color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

   8. +-commutative N/A

      \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}} + \frac{2 \cdot \mathsf{PI}\left(\right)}{3}\right) \]

   10. lift-/.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}}\right) \]

   11. frac-2neg N/A

       \[\leadsto 2 \cdot \cos \left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3} + \color{blue}{\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)}}\right) \]

   12. frac-add N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}{3 \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)} \]

   13. div-inv N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{3 \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \left(\mathsf{neg}\left(3\right)\right) + 3 \cdot \left(\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \frac{1}{3 \cdot \left(\mathsf{neg}\left(3\right)\right)}\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\cos^{-1} \left(\frac{g}{-h}\right), -3, \pi \cdot -6\right) \cdot -0.1111111111111111\right)} \]

5. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(\mathsf{fma}\left(\cos^{-1} \left(-\frac{g}{h}\right), -3, \pi \cdot -6\right) \cdot -0.1111111111111111\right) \]
6. Add Preprocessing

Alternative 6: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(2, \pi, \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \end{array} \]

FPCore

(FPCore (g h)
 :precision binary64
 (* 2.0 (cos (* 0.3333333333333333 (fma 2.0 PI (acos (- (/ g h))))))))

C

double code(double g, double h) {
	return 2.0 * cos((0.3333333333333333 * fma(2.0, ((double) M_PI), acos(-(g / h)))));
}

Julia

function code(g, h)
	return Float64(2.0 * cos(Float64(0.3333333333333333 * fma(2.0, pi, acos(Float64(-Float64(g / h)))))))
end

Wolfram

code[g_, h_] := N[(2.0 * N[Cos[N[(0.3333333333333333 * N[(2.0 * Pi + N[ArcCos[(-N[(g / h), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{2 \cdot \color{blue}{\mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\color{blue}{2 \cdot \mathsf{PI}\left(\right)}}{3} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   3. frac-2neg N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)}} + \frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}\right) \]

   4. lift-neg.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)} + \frac{\cos^{-1} \left(\frac{\color{blue}{\mathsf{neg}\left(g\right)}}{h}\right)}{3}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)} + \frac{\cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   6. lift-acos.f64 N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)} + \frac{\color{blue}{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}}{3}\right) \]

   7. frac-2neg N/A

      \[\leadsto 2 \cdot \cos \left(\frac{\mathsf{neg}\left(2 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{neg}\left(3\right)} + \color{blue}{\frac{\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)}{\mathsf{neg}\left(3\right)}}\right) \]

   8. frac-2neg N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{2 \cdot \mathsf{PI}\left(\right)}{3}} + \frac{\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)}{\mathsf{neg}\left(3\right)}\right) \]

   9. div-inv N/A

      \[\leadsto 2 \cdot \cos \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3}} + \frac{\mathsf{neg}\left(\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)}{\mathsf{neg}\left(3\right)}\right) \]

   10. frac-2neg N/A

       \[\leadsto 2 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3} + \color{blue}{\frac{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}{3}}\right) \]

   11. div-inv N/A

       \[\leadsto 2 \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{3} + \color{blue}{\cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right) \cdot \frac{1}{3}}\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{3} \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto 2 \cdot \cos \color{blue}{\left(\frac{1}{3} \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto 2 \cdot \cos \left(\color{blue}{\frac{1}{3}} \cdot \left(2 \cdot \mathsf{PI}\left(\right) + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   15. lift-*.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \left(\color{blue}{2 \cdot \mathsf{PI}\left(\right)} + \cos^{-1} \left(\frac{\mathsf{neg}\left(g\right)}{h}\right)\right)\right) \]

   16. lower-fma.f64 98.5

       \[\leadsto 2 \cdot \cos \left(0.3333333333333333 \cdot \color{blue}{\mathsf{fma}\left(2, \pi, \cos^{-1} \left(\frac{-g}{h}\right)\right)}\right) \]

   17. lift-/.f64 N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(g\right)}{h}\right)}\right)\right) \]

   18. frac-2neg N/A

       \[\leadsto 2 \cdot \cos \left(\frac{1}{3} \cdot \mathsf{fma}\left(2, \mathsf{PI}\left(\right), \cos^{-1} \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(g\right)\right)\right)}{\mathsf{neg}\left(h\right)}\right)}\right)\right) \]

4. Applied rewrites 98.5%

   \[\leadsto 2 \cdot \cos \color{blue}{\left(0.3333333333333333 \cdot \mathsf{fma}\left(2, \pi, \cos^{-1} \left(\frac{g}{-h}\right)\right)\right)} \]

5. Final simplification 98.5%

   \[\leadsto 2 \cdot \cos \left(0.3333333333333333 \cdot \mathsf{fma}\left(2, \pi, \cos^{-1} \left(-\frac{g}{h}\right)\right)\right) \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))